U.S. patent number 11,187,649 [Application Number 16/317,007] was granted by the patent office on 2021-11-30 for method for conducting optical measurement usingfull mueller matrix ellipsometer.
This patent grant is currently assigned to AK OPTICS TECHNOLOGY CO., LTD.. The grantee listed for this patent is AK OPTICS TECHNOLOGY CO., LTD.. Invention is credited to Gaozeng Cui, Guoguang Li, Tao Liu, Langfeng Wen, Wei Xiong.
United States Patent |
11,187,649 |
Liu , et al. |
November 30, 2021 |
Method for conducting optical measurement usingfull Mueller matrix
ellipsometer
Abstract
A method for conducting optical measurement using a full Mueller
matrix ellipsometer, which belongs to the technical field of
optical measurements. The optical measurement method comprises:
constructing an experimental optical path of a full Mueller matrix
ellipsometer; conducting complete regression calibration on the
full Mueller matrix ellipsometer; placing a sample to be measured
on a sample platform, and obtaining an experimental Fourier
coefficient of the sample to be measured; and according to the
experimental Fourier coefficient of the sample to be measured,
obtaining information about the sample to be measured. Since a
calibration method for the full Mueller matrix ellipsometer is not
only simple in operation process, but also makes full use of data
of the full Mueller matrix ellipsometer measured at the same time,
the introduced error is relatively small and the parameter obtained
by calibration is more accurate, so that the measurement result is
more accurate when the sample to be measured is measured. Thus, the
process of the optical measurement method is simplified.
Inventors: |
Liu; Tao (Beijing,
CN), Cui; Gaozeng (Beijing, CN), Li;
Guoguang (Beijing, CN), Xiong; Wei (Beijing,
CN), Wen; Langfeng (Beijing, CN) |
Applicant: |
Name |
City |
State |
Country |
Type |
AK OPTICS TECHNOLOGY CO., LTD. |
Beijing |
N/A |
CN |
|
|
Assignee: |
AK OPTICS TECHNOLOGY CO., LTD.
(Beijing, CN)
|
Family
ID: |
53198308 |
Appl.
No.: |
16/317,007 |
Filed: |
August 19, 2014 |
PCT
Filed: |
August 19, 2014 |
PCT No.: |
PCT/CN2014/084683 |
371(c)(1),(2),(4) Date: |
January 10, 2019 |
PCT
Pub. No.: |
WO2015/078202 |
PCT
Pub. Date: |
June 04, 2015 |
Prior Publication Data
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Document
Identifier |
Publication Date |
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US 20190317010 A1 |
Oct 17, 2019 |
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Foreign Application Priority Data
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Nov 26, 2013 [CN] |
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201310611422.8 |
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Current U.S.
Class: |
1/1 |
Current CPC
Class: |
G01N
21/211 (20130101); G01N 21/274 (20130101); G01N
2201/127 (20130101); G01N 2021/213 (20130101) |
Current International
Class: |
G01N
21/21 (20060101); G01N 21/27 (20060101) |
Field of
Search: |
;702/85 |
References Cited
[Referenced By]
U.S. Patent Documents
Foreign Patent Documents
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102878940 |
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Jan 2013 |
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CN |
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103134592 |
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Jun 2013 |
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CN |
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103163077 |
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Jun 2013 |
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CN |
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Other References
An office action in relation to Chinese patent application No.
201310611422.8. cited by applicant.
|
Primary Examiner: Chowdhury; Tarifur R
Assistant Examiner: Nixon; Omar H
Attorney, Agent or Firm: Platinum Intellectual Property
LLP
Claims
The invention claimed is:
1. A method for conducting optical measurement with a full Mueller
matrix ellipsometer, comprising the following steps: constructing
an experimental optical path of the full Mueller matrix
ellipsometer, the experimental optical path of the full Mueller
matrix ellipsometer including a light source, a polarizer, a first
phase compensator, an analyzer, a second phase compensator, a
spectrometer, and a sample stage; performing a total regression
calibration on operating parameters of the full Mueller matrix
ellipsometer; placing a sample to be tested on the sample stage,
and obtaining experimental Fourier coefficients of the sample to be
tested with the full Mueller matrix ellipsometer; obtaining
information of the sample to be tested based on the experimental
Fourier coefficients of the sample to be tested; wherein a method
for calibrating the full Mueller matrix ellipsometer comprises the
following steps: setting rotational speeds of the first phase
compensator and the second phase compensator; setting a frequency
of the spectrometer for measuring light intensity data, so that the
spectrometer measures the light intensity data every T/N time,
wherein a total of N sets of light intensity data are acquired,
where N.gtoreq.25, and T is a period of measurement; acquiring the
light intensity data measured by the spectrometer; obtaining
respective experimental Fourier coefficients .alpha.'.sub.2n,
.beta.'.sub.2n from N relation formulas between the light intensity
data and experimental Fourier coefficients formed by the N sets of
light intensity data, based on the light intensity data acquired by
a data acquisition module of the spectrometer; obtaining respective
theoretical Fourier coefficients .alpha..sub.2n, .beta..sub.2n
based on the respective experimental Fourier coefficients, an
initial polarization angle C.sub.s1 of the first phase compensator
and an initial polarization angle C.sub.s2 of the second phase
compensator which have been calibrated; obtaining, by a phase
retardation calculation module for the first phase compensator, a
phase retardation .delta..sub.1 of the first phase compensator
based on the respective theoretical Fourier coefficients, a
polarization angle P.sub.s of the polarizer and a polarization
angle A.sub.s of the analyzer which have been calibrated, on the
basis that a reference sample is isotropic and uniform; obtaining,
by a phase retardation calculation module for the second phase
compensator, a phase retardation .delta..sub.2 of the second phase
compensator, based on the respective theoretical Fourier
coefficients, the polarization angle P.sub.s of the polarizer and
the polarization angle A.sub.s of the analyzer which have been
calibrated, on the basis that the reference sample is isotropic and
uniform; obtaining accurate values of all operating parameters (d,
.theta., P.sub.s, A.sub.s, C.sub.s1, C.sub.s2, .delta..sub.1,
.delta..sub.2) of the full Mueller matrix ellipsometer through
least square fitting according to the relation formulas between the
theoretical Fourier coefficients and the operating parameters, with
(d, .theta., P.sub.s, A.sub.s, C.sub.s1, C.sub.s2, .delta..sub.1,
.delta..sub.2) being as variables, and with the initial
polarization angle C.sub.s1 of the first phase compensator, the
initial polarization angle C.sub.s2 of the second phase
compensator, the polarization angle P.sub.s of the polarizer, the
polarization angle A.sub.s of the analyzer, the phase retardation
.delta..sub.1 of the first phase compensator and the phase
retardation .delta..sub.2 of the second phase compensator, which
have been calibrated, being as initial values, where d is a
thickness of the reference sample, and .theta. is an angle at which
light is incident on the reference sample.
2. The method of claim 1, further comprising the following steps:
obtaining respective
.theta..sub.2n=tan.sup.-1(.beta.'.sub.2n/.alpha.'.sub.2n) based on
the respective experimental Fourier coefficients .alpha.'.sub.2n,
.beta.'.sub.2n; obtaining the initial polarization angle C.sub.s1
of the first phase compensator based on the respective
.theta..sub.2n; obtaining the initial polarization angle C.sub.s2
of the second phase compensator based on the respective
.theta..sub.2n; obtaining the polarization angle P.sub.s of the
polarizer based on the respective .theta..sub.2n; and obtaining the
polarization angle A.sub.s of the analyzer based on the respective
.theta..sub.2n.
3. The method of claim 1, wherein N=25, and an experimental Fourier
coefficient calculation module directly obtains the respective
experimental Fourier coefficients .alpha.'.sub.2n, .beta.'.sub.2n
based on N relation formulas between the light intensity data and
experimental Fourier coefficients formed by the N sets of light
intensity data.
4. The method of claim 1, wherein N>25, and an experimental
Fourier coefficient calculation module obtains the respective
experimental Fourier coefficients .alpha.'.sub.2n, .beta.'.sub.2n
through least square method based on N relation formulas between
the light intensity data and experimental Fourier coefficients
formed by the N set of light intensity data.
5. The method of claim 1, wherein the light source is a broad
spectrum light source, and the number of wavelengths of light which
can be generated by the light source is N', and the number of
relation formulas between the theoretical Fourier coefficients and
the operating parameters is 24.times.N'.
6. The method of claim 5, wherein the number of the reference
samples which are isotropic and uniform is m, and the number of
relation formulas between the theoretical Fourier coefficients and
the operating parameters is 24.times.N'.times.m.
7. The method of claim 1, wherein the reference sample is a silicon
dioxide film with silicon as a substrate.
Description
TECHNICAL FIELD
The present disclosure relates to the field of optical measurement
technology, and more particularly to a method for conducting
optical measurement with a full Mueller matrix ellipsometer.
BACKGROUND OF THE INVENTION
An ellipsometer is an optical measuring instrument that takes
advantage of the polarization characteristics of light to acquire
information of a sample to be tested. The working principle of the
ellipsometer is as below: letting light passing through a polarizer
be incident on a sample to be tested; obtaining the information of
the sample to be tested by measuring a change of polarization state
(amplitude ratio and phase difference) of an incident light and a
reflected light on a surface of the sample to be tested. The
ellipsometer with rotatory polarizer or single rotatory compensator
can obtain up to 12 parameters of the sample in one measurement.
With the advancement of the integrated circuit technology and the
complexity of device structure, unknown variables to be measured
are continuously increased, and traditional ellipsometers present
certain limitations in various aspects, such as film thickness
measurement of ultra-thin films, measurement of optical constants
for anisotropic materials, depolarization analysis of surface
features, and measurement of critical dimensions and topography in
integrated circuits. A full Mueller matrix ellipsometer
(ellipsometer in a broad sense) can acquire 16 parameters of
4.times.4 order Mueller matrix in one measurement, obtaining more
abundant information as compared with a traditional ellipsometer.
It breaks through technical limitations of traditional
ellipsometers and enables accurate, fast, non-destructive
measurement of film thickness, optical constants, critical
dimensions and three-dimensional topography in a wide spectral
range.
The key to ensure measurement accuracy and maintain device status
for a spectroscopic ellipsometer is the calibration of the device.
The ellipsometer may generate system deviation gradually during use
as time goes on, especially a thickness of a wave plate is
susceptible to changes in temperature and pressure as well as
environmental deliquescence. Therefore, a calibration method
enabling quick and accurate correction of the ellipsometer is a key
technique to ensure device effectiveness and production efficiency.
With a calibration process of an existing conventional ellipsometer
(FIG. 1), as indicated in Chinese Patent No. 201210375771.X, when a
polarization direction of a polarizer is calibrated, typically, a
position P1 of the polarizer is fixed in the vicinity of 0.degree.,
and then an analyzer A is rotated and a light intensity I.sub.1 is
measured so as to obtain an I.sub.1(t) curve in this state; then an
angle of the polarizer P is changed to put the polarizer P at a
position P2, and then the light intensity I.sub.2 is measured so as
to obtain an I.sub.2(t) curve. The above steps are repeated to
measure the light intensities when the polarizer P is at different
angles so as to obtain I(t) curves when the polarizer P is at
different angles. Fourier series expansion is performed on the I(t)
curves above respectively to obtain Fourier coefficients of the
polarizer P at different angles; a function is established which is
associated with the Fourier coefficients and has a minimum value
when the polarization angle of the polarizer P is zero; through
data analysis, the position of the polarizer P where the function
has the minimum value is found. It can be considered that the angle
of the polarizer P is 0 at this position (see Spectroscopic
Ellipsometry Principles and Applications, Hiroyuki Fujiwara, 2007
for details). Then, a value of the polarization direction As of the
analyzer at a start position is calculated through the Fourier
coefficients. With this calibration method, not only the rotation
of the analyzer but also the electric or manual rotation of the
polarizer P is required, that is, after the polarization direction
of the polarizer is determined, the angle of the polarizer needs to
be manually or electrically adjusted. In this case, due to
instability of mechanical structure and/or error in human
operation, an error between an actual angle and an angle desired to
be set may be caused, which easily leads to inaccuracy in the
measurement of a reference sample. Therefore, if this method is
used, the accuracy in angle calibration of the polarizer is
relatively low, and thus the measurement accuracy of the
ellipsometer is limited. The angle of incidence of light in an
ellipsometer can be obtained by manual measurement, but the manual
measurement has limited accuracy, and it is sometimes required to
measure a reference sample at different angles of incidence to
obtain more information thereof, the manual measurement is easy to
lead to a wrong result of data analysis due to artificial
adjustment error or reading error. Chinese Patent No.
201010137774.0 discloses a device for automatically detecting an
angle of incidence in an ellipsometry system, which can realize
automatic detection of the angle of incidence, but the device
requires position detecting devices to be installed at several
places in the system, which makes systematic structure complicated.
Moreover, the calibration of the position detecting devices itself
is also a complicated process, thus the application of such an
automatic detecting device in an ellipsometer is limited.
In a systematic calibration of an existing full Mueller matrix
ellipsometer, such as the Mueller ellipsometer in US Patent
US005956147, a photoelastic modulator (PEM) is used as a phase
compensator. When a phase retardation of the PEM is calibrated, it
is built in a straight-through ellipsometry system for measurement,
and the PEM needs to be taken off the original equipment to measure
its corresponding phase retardation. After the calibration is
completed, the PEM is reloaded onto the equipment. During the
mechanical loading and unloading processes, it cannot be guaranteed
that loading position is the same as the previous loading position,
which increases systematic error, and re-construction of the
straight-through measuring system will increase workload. In the
existing literature (Harland G. Tompkins, Eugene A. Irene, Handbook
of ellipsometry, 7.3.3.4 Calibration 7), a Mueller ellipsometer
uses a wave plate as a phase compensator, the process of which is
to build a straight-through measuring platform on an experimental
table to measure Fourier coefficients obtained experimentally and
use .delta..sub.1=
.times..times.''.times..times..times..times..delta..times..times.''
##EQU00001## where |B'.sub.B|= {square root over
((.alpha.'.sub.2n).sup.2+(.beta.'.sub.2n).sup.2)} for calibration.
It is required to remove two phase compensators during calibration
and then put back, which increases systematic error. If the
calibration is carried out without removing the phase compensators,
obliquely-incident measuring arms on both sides of the sample must
be rotated to a horizontal position (eg. Woollam's ellipsometer as
shown in FIG. 3, the incident arm is rotated from position 1 to
position 3 during calibration, and the exiting arm is rotated from
position 2 to position 4), which increases the complexity of the
system.
In summary, with current techniques, delay spectral lines of all
phase compensators being used must be tested prior to device
assembly, and a phase retardation of the phase compensator must be
calibrated using a straight-through ellipsometry system. The system
is required to have a design to adjust an angle of incidence to a
straight-through type, and there is a process of changing the angle
of incidence during the calibration process. These methods increase
the complexity of the system and the calibration process is more
complicated.
Since a method for conducting optical measurement with a full
Mueller matrix ellipsometer is performed after the calibration of
the full Mueller matrix ellipsometer, the complexity in the
calibration process of the full Mueller matrix ellipsometer must
result in the complexity of the method for conducting optical
measurement with a full Mueller matrix ellipsometer.
SUMMARY OF THE INVENTION
In order to solve the above problems, the present disclosure
proposes a simplified method for conducting optical measurement
with a full Mueller matrix ellipsometer whose calibration process
is simplified.
A method for conducting optical measurement with a full Mueller
matrix ellipsometer provided by the present disclosure may comprise
the following steps:
constructing an experimental optical path of the full Mueller
matrix ellipsometer, and wherein the experimental optical path of
the full Mueller matrix ellipsometer includes a light source, a
polarizer, a first phase compensator, an analyzer, a second phase
compensator, a spectrometer, and a sample stage;
performing a total regression calibration on the full Mueller
matrix ellipsometer;
placing a sample to be tested on the sample stage, and obtaining
experimental Fourier coefficients of the sample to be tested with
the full Mueller matrix ellipsometer;
obtaining information of the sample to be tested based on the
experimental Fourier coefficients of the sample to be tested.
In addition, a method for calibrating the full Mueller matrix
ellipsometer may comprise the following steps:
setting rotational speeds of the first phase compensator and the
second phase compensator;
setting a frequency of the spectrometer for measuring light
intensity data, so that the spectrometer measures the light
intensity data every T/N time, wherein a total of N sets of light
intensity data are acquired, where N.gtoreq.25, and T is a period
of measurement;
acquiring the light intensity data measured by the
spectrometer;
obtaining respective experimental Fourier coefficients
.alpha.'.sub.2n, .beta.'.sub.2n from N relation formulas between
the light intensity data and experimental Fourier coefficients
formed by the N sets of light intensity data, based on the light
intensity data acquired by a data acquisition module of the
spectrometer;
obtaining respective theoretical Fourier coefficients
.alpha..sub.2n, .beta..sub.2n based on the respective experimental
Fourier coefficients, an initial polarization angle C.sub.s1 of the
first phase compensator and an initial polarization angle C.sub.s2
of the second phase compensator which have been calibrated;
obtaining, by a phase retardation calculation module for the first
phase compensator, a phase retardation .delta..sub.1 of the first
phase compensator based on the respective theoretical Fourier
coefficients, a polarization angle P.sub.s of the polarizer and a
polarization angle A.sub.s of the analyzer which have been
calibrated, on the basis that a reference sample is isotropic and
uniform;
obtaining, by a phase retardation calculation module for the second
phase compensator, a phase retardation .delta..sub.2 of the second
phase compensator, based on the respective theoretical Fourier
coefficients, the polarization angle P.sub.s of the polarizer and
the polarization angle A.sub.s of the analyzer which have been
calibrated, on the basis that the reference sample is isotropic and
uniform;
obtaining accurate values of all operating parameters (d, .theta.,
P.sub.s, A.sub.s, C.sub.s1, C.sub.s2, .delta..sub.1, .delta..sub.2)
of the full Mueller matrix ellipsometer through least square
fitting according to the relation formulas between the theoretical
Fourier coefficients and the operating parameters, with (d,
.theta., P.sub.s, A.sub.s, C.sub.s1, C.sub.s2, .delta..sub.1,
.delta..sub.2) being as variables, and with the initial
polarization angle C.sub.s1 of the first phase compensator, the
initial polarization angle C.sub.s2 of the second phase
compensator, the polarization angle P.sub.s of the polarizer, the
polarization angle A.sub.s of the analyzer, the phase retardation
.delta..sub.1 of the first phase compensator and the phase
retardation .delta..sub.2 of the second phase compensator, which
have been calibrated, being as initial values, where d is a
thickness of the reference sample, and .theta. is an angle at which
light is incident on the reference sample.
The method for conducting optical measurement with a full Mueller
matrix ellipsometer according to the present disclosure may utilize
a reference sample which is isotropic and uniform, and obtain the
phase retardation .delta..sub.1 of the first phase compensator and
the phase retardation .delta..sub.2 of the second phase compensator
based on the relation formulas between the light intensity data and
the experimental Fourier coefficients as well as the polarization
angle P.sub.s of the polarizer and the polarization angle A.sub.s
of the analyzer which have been calibrated; and then obtains
accurate values of all operating parameters (d, .theta., P.sub.s,
A.sub.s, C.sub.s1, C.sub.s2, .delta..sub.1, .delta..sub.2) of the
full Mueller matrix ellipsometer by least square fitting according
to the relation formulas between the theoretical Fourier
coefficients and the operating parameters, with (d, .theta.,
P.sub.s, A.sub.s, C.sub.s1, C.sub.s2, .delta..sub.1, .delta..sub.2)
being as variables, and with the initial polarization angle
C.sub.s1 of the first phase compensator, the initial polarization
angle C.sub.s2 of the second phase compensator, the polarization
angle P.sub.s of the polarizer, the polarization angle As of the
analyzer, the phase retardation .delta..sub.1 of the first phase
compensator, the phase retardation .delta..sub.2 of the second
phase compensator, which have been calibrated, being as initial
values. The calibration method can take full advantages of
measurement data obtained at a same time, which introduces
relatively small error and obtains more accurate parameters after
calibration. Thus, the result of measurement is more accurate when
a sample to be tested is measured using the method of the present
disclosure.
BRIEF DESCRIPTION OF THE DRAWINGS
FIG. 1 is a diagram showing an experimental optical path of a full
Mueller matrix ellipsometer constructed in a method for conducting
optical measurement with a full Mueller matrix ellipsometer
according to an embodiment of the present invention;
FIG. 2 is a logic block diagram of a method for conducting optical
measurement with a full Mueller matrix ellipsometer according to an
embodiment 1 of the present invention; and
FIG. 3 is a logic block diagram of a method for conducting optical
measurement with a full Mueller matrix ellipsometer according to an
embodiment 2 of the present invention.
DETAILED DESCRIPTION OF THE INVENTION
The present invention will be described in detail below in
conjunction with the drawings and specific embodiments for the
in-depth understanding of the invention.
Embodiment 1
A method for conducting optical measurement with a full Mueller
matrix ellipsometer according to Embodiment 1 of the present
invention may comprise the following steps:
step 1: referring to FIG. 1, constructing an experimental optical
path of a full Mueller matrix ellipsometer. The experimental
optical path may include a light source 1, an annular mirror 2, a
pinhole 3, a first off-axis parabolic mirror 4, a polarizer 5, a
first phase compensator 6, a first plane mirror 7, a sample stage
8, a second off-axis parabolic mirror 9, a third off-axis parabolic
mirror 10, a second plane mirror 11, a second phase compensator 12,
an analyzer 13, a fourth off-axis parabolic mirror 14, a
spectrometer 15 and a terminal 16. An isotropic and uniform
reference sample is carried on the sample stage 8. The experimental
optical path of the full Mueller matrix ellipsometer, which can be
self-calibrated through total regression, may have the following
optical process:
S.sub.out=M.sub.AR(A')R(-C.sub.2)M.sub.c2(.delta..sub.2)R(C'.sub.2).times-
.M.sub.s.times.R(-C'.sub.1)M.sub.c1(.delta..sub.1)R(C'.sub.1)R(-P')M.sub.p-
R(P)S.sub.in that is,
.function..times..times..times..times..times..times..times..times..times.-
.times..times..times..times.
.times..times.'.times..times.'.times..times.'.times..times..times.'.funct-
ion..times..times..delta..times..times..delta..times..times..delta..times.-
.times..delta..times.
.times..times.'.times..times.'.times..times.'.times..times..times.'.funct-
ion..times.
.times..times.'.times..times.'.times..times.'.times..times..times.'.funct-
ion..times..times..delta..times..times..delta..times..times..delta..times.-
.times..delta..times.
.times..times.'.times..times.'.times..times.'.times..times..times.'.funct-
ion..times..times..times..times..times..times..times..times..times..times.-
.times..times..times.
.function..times..times..times..times..times..times..times..times..times.-
.times..times..times..function. ##EQU00002##
Step 2 of the method may comprise: performing a total regression
calibration on the full Mueller matrix ellipsometer.
Step 3 of the method may comprise: placing the sample to be tested
on the sample stage, and obtaining experimental Fourier
coefficients of the sample to be tested with the full Mueller
matrix ellipsometer.
Step 4 of the method may comprise: obtaining information of the
sample to be tested based on the experimental Fourier coefficients
of the sample to be tested.
The experimental Fourier coefficients have relation to Mueller
elements of the sample, an azimuth angle P.sub.s of the polarizer,
an azimuth angle A.sub.s of the analyzer, azimuth angles C.sub.s1
and C.sub.s2 of the two phase compensators, and phase retardations
.delta..sub.1 and .delta..sub.2 (refer to Harland G. Tompkins,
Eugene A. Irene, Handbook of ellipsometry, 7.3.3 Dual Rotating
Compensator 7). However, the Mueller elements of the sample are
related to optical constants n, k of a material of the sample, a
thickness d, an angle .theta. at which light beams are incident on
the sample, and a wavelength .lamda. of the light beams. Therefore,
after the experimental Fourier coefficients of the sample are
measured, the Mueller elements of the sample can be obtained
according to the above relationships, and then the information of
the sample can be obtained.
In an embodiment, the process of performing a total regression
calibration on the full Mueller matrix ellipsometer comprises the
following steps:
step 21: setting rotational speeds of the first and second phase
compensators;
step 22: setting a frequency of a spectrometer for measuring light
intensity data, so that the spectrometer may measure the light
intensity data every T/N time, and a total of N sets of light
intensity data are acquired, wherein N.gtoreq.25, and T is a period
of measurement;
step 23: acquiring the light intensity data measured by the
spectrometer;
step 24: obtaining respective experimental Fourier coefficients
.alpha.'.sub.2n/.beta.'.sub.2n from N relation formulas between the
light intensity data and the experimental Fourier coefficients
formed by the N sets of light intensity data, based on the light
intensity data acquired by a data acquisition module of the
spectrometer;
step 25: obtaining respective theoretical Fourier coefficients
.alpha..sub.2n, .beta..sub.2n according to the respective
experimental Fourier coefficients, an initial polarization angle
C.sub.s1 of the first phase compensator and the initial
polarization angle C.sub.s2 of the second phase compensator which
have been calibrated;
step 26: obtaining, on the basis that a reference sample is
isotropic and uniform, by a phase retardation calculation module
for the first phase compensator, phase retardation .delta..sub.1 of
the first phase compensator based on the respective theoretical
Fourier coefficients, a polarization angle P.sub.s of the polarizer
and a polarization angle A.sub.s of the analyzer which have been
calibrated; and
obtaining, on the basis that the reference sample is isotropic and
uniform, by a phase retardation calculation module for the second
phase compensator, phase retardation .delta..sub.2 of the second
phase compensator based on the respective theoretical Fourier
coefficients, a polarization angle P.sub.s of the polarizer and a
polarization angle A.sub.s of the analyzer which have been
calibrated;
step 27: obtaining accurate values of all operating parameters (d,
.theta., P.sub.s, A.sub.s, C.sub.s1, C.sub.s2, .delta..sub.1,
.delta..sub.2) of the full Mueller matrix ellipsometer through
least square fitting according to the relation formulas between the
theoretical Fourier coefficients and the operating parameters, with
(d, .theta., P.sub.s, A.sub.s, C.sub.s1, C.sub.s2, .delta..sub.1,
.delta..sub.2) being as variables, and with the initial
polarization angle C.sub.s1 of the first phase compensator, the
initial polarization angle C.sub.s2 of the second phase
compensator, the polarization angle P.sub.s of the polarizer, the
polarization angle A.sub.s of the analyzer, the phase retardation
.delta..sub.1 of the first phase compensator, the phase retardation
.delta..sub.2 of the second phase compensator, which have been
calibrated, being as initial values, where d is a thickness of the
reference sample, and .theta. is an angle at which light is
incident on the reference sample.
A corresponding Mueller matrix of the reference sample that is
isotropic and uniform is:
##EQU00003##
Taking N=36, the ratio of the rotation speed of the first phase
compensator 6 to the rotation speed of the second phase compensator
12 equaling 5:3 as an example, at this time, the first phase
compensator 6 and the second phase compensator 12 are respectively
in a rotating state, and the ratio of the rotation speed of the
first phase compensator 6 to the rotation speed of the second phase
compensator 12 equals 5:3. At this time, C'.sub.1=5(C-C.sub.s1),
C'.sub.2=3(C-C.sub.s2), and the time during which the first phase
compensator 6 rotates 5 turns or the second phase compensator 12
rotates 3 turns is a period T, and where:
-C.sub.s1, an angle of a fast optical axis of the first phase
compensator 6 at a time t=0,
-C.sub.s2, an angle of a fast optical axis of the second phase
compensator 12 at a time t=0,
C=.omega.t, a rotation angle by which the first phase compensator 6
and the second phase compensator 12 rotate at a fundamental
physical frequency .omega..
.intg..times..times.'.function..times..alpha..times.'.times..times..times-
..times..times..times..omega..times..times..beta..times.'.times..times..ti-
mes..times..times..times..times..omega..times..times..times..pi..times..ti-
mes.'.times..omega..times.'.times..times..omega..times..times..times..time-
s..pi..function..alpha..times.'.times..times..times..times..times..times..-
pi..beta..times.'.times..times..times..times..times..times..pi..times..tim-
es..times..times..times. ##EQU00004##
where .omega.=.pi./T.
With the acquired S1, S2, S3 . . . S36, 36 equations containing 25
unknowns can be obtained through the above formula (n=9, 12, 14,
15, the primed Fourier coefficients .alpha.'.sub.2n=0 and
.beta.'.sub.2n=0). Through a nonlinear least square method, a total
of 24 primed Fourier coefficients .alpha.'.sub.2n and
.beta.'.sub.2n can be obtained.
The transformation relationship between the theoretical Fourier
coefficients .alpha..sub.2n and .beta..sub.2n and the experimental
Fourier coefficients .alpha.'.sub.2n and .beta.'.sub.2n is
expressed in formulas 2.7 and 2.8: .alpha..sub.2n=.alpha.'.sub.2n
cos .PHI..sub.2n+.beta.'.sub.2n sin .PHI..sub.2n 2.7
.beta..sub.2n=-.alpha.'.sub.2n sin .PHI..sub.2n+.beta.'.sub.2n cos
.PHI..sub.2n 2.8 where:
.PHI..sub.2=12C.sub.s2-10C.sub.s1;
.PHI..sub.4=10C.sub.s1-6C.sub.s2;
.PHI..sub.6=6C.sub.s2; .PHI..sub.8=20C.sub.s1-12C.sub.s2;
.PHI..sub.10=10C.sub.s1; .PHI..sub.12=12C.sub.s2;
.PHI..sub.14=20C.sub.s1-6C.sub.s2;
.PHI..sub.16=10C.sub.s1+6C.sub.s2;
.PHI..sub.20=20C.sub.s1; .PHI..sub.22=10C.sub.s1+12C.sub.s2;
.PHI..sub.26=20C.sub.s1+6C.sub.s2;
.PHI..sub.32=20C.sub.s1+12C.sub.s2
The theoretical Fourier coefficients .alpha..sub.2n and
.beta..sub.2n can be obtained from the formulas 2.7 and 2.8.
For a sample which is isotropic and uniform,
M.sub.13=M.sub.31=M.sub.14=M.sub.41=M.sub.23=M.sub.32=M.sub.24=M.sub.42=0-
. Further, according to theoretical principles of the Mueller
ellipsometer, the following theoretical expressions for the
theoretical Fourier coefficients .alpha..sub.2, .beta..sub.2,
.alpha..sub.10, .beta..sub.10, .alpha..sub.6, .beta..sub.6,
.alpha..sub.14, .beta..sub.14, .alpha..sub.22, .beta..sub.22,
.alpha..sub.26, .beta..sub.26 can be obtained:
.alpha..times..times..delta..times..times..times..delta..times..times..ti-
mes..times..times..times..times..times..beta..times..times..delta..times..-
times..times..delta..times..times..times..times..times..times..times..time-
s..alpha..times..times..times..delta..times..times..times..delta..times..t-
imes..times..times..beta..times..times..times..delta..times..times..times.-
.delta..times..times..times..times..alpha..times..times..times..delta..tim-
es..times..times..delta..times..times..times..times..beta..times..times..t-
imes..delta..times..times..times..delta..times..times..times..times.
##EQU00005##
From formulas 2.10 and 2.12, it can be obtained:
.beta..beta..times..times..times..times..times..times..times..times..time-
s..times..times..times..times..delta..times..times..delta..function..times-
..beta..times..times..times..times..times..times..times..times..beta..time-
s..times..times..times..times..beta..times..times..times..times..times..ti-
mes..times..times..beta..times..times..times..times..times..times..times..-
times..noteq..times..times..pi..times..times..times..times..noteq..times..-
times..pi..times..times..times..times..noteq..pi..times..times..pi.
##EQU00006## (n is an integer) (the Fourier coefficients must be
guaranteed to be non-zero)
Similarly, the phase retardation .delta..sub.1 of the compensator
can also be calculated through formulas 2.9 and 2.12, formulas 2.9
and 2.11, formulas 2.10 and 2.11, formulas 2.9 and 2.13, formulas
2.9 and 2.14, formulas 2.10 and 2.13, formulas 2.10 and 2.14.
The phase retardation .delta..sub.2 of the second compensator is
calibrated below.
.alpha..times..times..times..times..delta..times..times..delta..times..ti-
mes..times..times..beta..times..times..times..times..delta..times..times..-
delta..times..times..times..times..alpha..times..times..times..delta..time-
s..times..delta..times..times..times..times..times..times..times..times..b-
eta..times..times..times..delta..times..times..delta..times..times..times.-
.times..times..times..times..times..alpha..times..times..times..times..del-
ta..times..times..delta..times..times..times..times..beta..times..times..t-
imes..times..delta..times..times..delta..times..times..times..times.
##EQU00007##
From formulas 2.16 and 2.18, it can be obtained:
.alpha..alpha..times..times..times..times..times..times..times..times..ti-
mes..times..times..times..times..delta..times..times..delta..function..tim-
es..alpha..times..times..times..times..times..times..times..times..alpha..-
times..times..times..times..times..alpha..times..times..times..times..time-
s..times..times..times..alpha..times..times..times..times..times..times..t-
imes..times..noteq..pi..times..times..pi..times..times..times..times..note-
q..times..times..pi..times..times..times..times..noteq..times..times..pi.
##EQU00008## (n is an integer) (the Fourier coefficients must be
guaranteed to be non-zero)
Similarly, the phase retardation .delta..sub.2 of the compensator
can also be calibrated through formulas 2.16 and 2.19, formulas
2.17 and 2.18, formulas 2.17 and 2.19, formulas 2.18 and 2.20,
formulas 2.18 and 2.21, formulas 2.19 and 2.20, formulas 2.19 and
2.21.
The transformation of the experimental Fourier coefficients and the
theoretical Fourier coefficients can be realized by the formulas
2.7 and 2.8. Meanwhile, the theoretical Fourier coefficients have
relation to Mueller elements of the sample, an azimuth angle
P.sub.s of the polarizer, an azimuth angle A.sub.s of the analyzer,
azimuth angles C.sub.s1 and C.sub.s2 of the two phase compensators,
and the phase retardations .delta..sub.1 and .delta..sub.2 (refer
to Harland G. Tompkins, Eugene A. Irene, Handbook of ellipsometry,
7.3.3 Dual Rotating Compensator 7). Mueller elements of the sample
are related to optical constants n, k of a material of the sample,
thickness d, an angle .theta. at which light beams are incident on
the sample, and a wavelength .lamda. of the light beams. The
experimental Fourier coefficients .alpha.'.sub.2n and
.beta.'.sub.2n are related to (n, k, d, .theta., .lamda., P.sub.s,
A.sub.s, C.sub.s1, C.sub.s2, .delta..sub.1, .delta..sub.2), .theta.
is an angle at which light beams are incident on the sample. For a
reference sample with known optical constants n, k under a single
wavelength, there are 24 .alpha.'.sub.2n and .beta.'.sub.2n in
total obtained by measurement in an experiment, and correspondingly
24 equations can be obtained, which are related only to (d,
.theta., .lamda., P.sub.s, A.sub.s, C.sub.s1, C.sub.s2,
.delta..sub.1, .delta..sub.2). The P.sub.s, A.sub.s, C.sub.s1,
C.sub.s2, .delta..sub.1, .delta..sub.2 obtained above by
calibration can be used as initial values, and the wavelength
corresponding to the measurement in the experiment are known; 24
equations obtained according to experimental Fourier coefficients
have relation to (d, .theta., P.sub.s, A.sub.s, C.sub.s1, C.sub.s2,
.delta..sub.1, .delta..sub.2). Thus, the remaining operating
parameters (d, .theta., P.sub.s, A.sub.s, C.sub.s1, C.sub.s2,
.delta..sub.1, .delta..sub.2) of the Mueller ellipsometer can be
obtained by the least square fitting. The reference sample may be a
silicon dioxide film sample with silicon as the substrate, the
optical constants n and k of which can be consulted in the
literatures. Taking the wavelength of 632.8 nm as an example, the
optical constants of the reference sample are n=1.457, and k=0.
When N=25, an experimental Fourier coefficient calculation module
directly obtains the respective experimental Fourier coefficients
.alpha.'.sub.2n, .beta.'.sub.2n based on N relation formulas
between light intensity data and experimental Fourier coefficients
formed by the N sets of light intensity data.
When N>25, the experimental Fourier coefficient calculation
module obtains the respective experimental Fourier coefficients
.alpha.'.sub.2n, .beta.'.sub.2n by the least square method
according to N relation formulas between light intensity data and
experimental Fourier coefficients formed by the N sets of light
intensity data.
The light source may be a broad spectrum light source. The number
of wavelengths of light which can be generated by the light source
is N', and the number of relation formulas between theoretical
Fourier coefficients and operating parameters may be
24.times.N'.
The number of the reference samples which are isotropic and uniform
may be m, and the number of relation formulas between theoretical
Fourier coefficients and operating parameters may be
24.times.N'.times.m.
Embodiment 2
Referring to FIG. 3, the difference between the total regression
self-calibrated full Mueller matrix ellipsometer according to the
embodiment 2 of the present invention and the total regression
self-calibrated full Mueller matrix ellipsometer according to the
embodiment 1 of the present invention lies in that: the total
regression self-calibration method of the total regression
self-calibrated full Mueller matrix ellipsometer according to the
embodiment 2 of the present invention may further comprise the
following steps:
obtaining respective .theta..sub.2n based on respective
experimental Fourier coefficients .alpha.'.sub.2n, .beta.'.sub.2n,
where .theta..sub.2n is an intermediate parameter defined for the
convenience of calculation;
obtaining an initial polarization angle C.sub.s1 of a first phase
compensator based on the respective .theta..sub.2n;
obtaining an initial polarization angle C.sub.s2 of a second phase
compensator based on the respective .theta..sub.2n;
obtaining a polarization angle P.sub.s of a polarizer based on the
respective .theta..sub.2n;
obtaining a polarization angle A.sub.s of an analyzer based on the
respective .theta..sub.2n;
where .theta..sub.2n=tan.sup.-1(.beta.'.sub.2n/.alpha.'.sub.2n)
2.2
Using the method available in the literature (R. W. Collins and
JoohyunKoh Dual rotating-compensator multichannel ellipsometer:
instrument design for real-time Mueller matrix spectroscopy of
surfaces and films Vol. 16, No. 8/August 1999/J. Opt. Soc. Am. A
1997 to 2006), which corresponds to the following formulas 2.3 to
2.6, the initial polarization angles Cs1 and Cs2 of the
compensators, as well as the polarization angles P.sub.s and
A.sub.s of the polarizer and the analyzer can be calibrated.
.times..times..theta..theta..times..times..theta..theta..theta..theta..ti-
mes..times..times..theta..theta..times..times..times.
##EQU00009##
On the basis of calibrated P.sub.s, A.sub.s, C.sub.s1, and
C.sub.s2, in the case that the compensators are not disassembled
from the experimental stage or equipment for separate measurement,
the method we proposed can calibrate phase retardations of both
compensators under different wavelengths in one experiment. The
calibration process is accurate and simple.
The above embodiments describe the objects, the technical solutions
and advantages of the present invention in detail. However, it
should be appreciated that the foregoing is only specific
embodiments of the present invention rather than limiting the
invention. Therefore, any modification, equivalent substitution,
improvement, etc. made within the spirit and principle of the
present invention are intended to be included within the scope of
the present invention.
* * * * *